feat(RingTheory/GradedAlgebra): homogeneous relation

[xyzw12345]

In this PR, we defined the concept of a homogeneous relation and proved some properties about homogeneous relations. The main result of this PR is showing that taking RingQuot by a homogeneous relation can give a graded structure on the quotient ring. This result can be used to define graded structures on rings obtained using RingQuot, e.g. the Symmetric Algebra defined in #21539 can be verified to have such a structure.

Co-authored-by:
Yiming Fu @pelicanhere [email protected]
Zhenyan Fu @pumpkin678 [email protected]
Raphael Douglas Giles @Raph-DG [email protected]
Jiedong Jiang @jjdishere [email protected]

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[github-actions]

PR summary 0dcce80bb1

Import changes for modified files

No significant changes to the import graph

Import changes for all files

Files	Import difference
Mathlib.RingTheory.GradedAlgebra.HomogeneousRelation (new file)	1024

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Declarations diff

+ Finset.relation_sum_induction
+ IsHomogeneousRelation
+ coeRingEquiv
+ coe_mul_sum_support_subset
+ eqvGen_proj_mul_left
+ eqvGen_proj_mul_right
+ eqvGen_ringQuot_add_right
+ eqvGen_ringQuot_mul_left
+ eqvGen_ringQuot_mul_right
+ eqvGen_ringQuot_of_eqvGen
+ gradedRing
+ instance (priority := 50) SubgroupClass.toSubgroup {A M : Type*} [Group M] [SetLike A M]
+ instance (priority := 50) SubmonoidClass.toSubmonoid {A M : Type*} [Monoid M] [SetLike A M]
+ instance : GradedAlgebra ((Submodule.map (RingQuot.mkAlgHom R rel)).comp 𝒜) := by
+ instance : GradedRing ((AddSubmonoid.map (RingQuot.mkRingHom rel)).comp 𝒜) := by
+ instance : IsHomogeneousRelation 𝒜 (Relation.EqvGen rel) := by
+ instance : IsHomogeneousRelation 𝒜 (RingConGen.Rel rel)
+ instance : IsHomogeneousRelation 𝒜 (RingQuot.Rel rel) := ⟨by
+ instance : SetLike.GradedMonoid ((AddSubmonoid.map (RingQuot.mkRingHom rel)).comp 𝒜)
+ instance : SetLike.GradedMonoid ((Submodule.map (RingQuot.mkAlgHom R rel)).comp 𝒜)
+ map
+ map_apply
+ map_eq_iff
+ map_of
+ map_surjective
+ mkAlgHom_eq_iff
+ mkAlgHom_eq_iff'
+ mkRingHom_eq_iff
+ mkRingHom_eq_iff'

You can run this locally as follows

## summary with just the declaration names:
./scripts/declarations_diff.sh <optional_commit>

## more verbose report:
./scripts/declarations_diff.sh long <optional_commit>

The doc-module for script/declarations_diff.sh contains some details about this script.

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No changes to technical debt.

You can run this locally as

./scripts/technical-debt-metrics.sh pr_summary

• The relative value is the weighted sum of the differences with weight given by the inverse of the current value of the statistic.
• The absolute value is the relative value divided by the total sum of the inverses of the current values (i.e. the weighted average of the differences).